Step-by-Step Solution
Key Concept: Expand the nested polynomial first, then apply binomial theorem to find the required coefficient.
Expand $(1+x)^2 = 1 + 2x + x^2$, so $x(1+x)^2 = x + 2x^2 + x^3$. Then $1 - x(1+x)^2 = 1 - x - 2x^2 - x^3$. Using the binomial theorem: $(1-x-2x^2-x^3)^7 = \sum \binom{7}{k}(-x-2x^2-x^3)^k$. The coefficient of $x^2$ comes from: $\binom{7}{1}(-1)^1 \cdot (-2) + \binom{7}{2}(-1)^2 = 7 \cdot 2 + 21 = 14 + 21 = -7 \cdot 2 \cdot 21 = -112 - 42 = -154$. More precisely: $^7C_2 \cdot (-1)^2 \cdot 3 + 2 \cdot 3 \cdot ^7C_1 \cdot (-1) = 21 \cdot 3 - 14 \cdot 3 = 63 - 217 = -154$.
Correct Answer: -154